Monday, June 1, 2009

Numbers That Have No Meaning

The Planck-time - the smallest slice of elapsed time that we can currently conceive of as physically meaningful - is about 5 x 10^-44 seconds. A year is 31,557,600 seconds long, and the universe is about 1.4 x 10^10 years old. This means that since the Big Bang, there have been about 8.8 x 10^60 Planck-times so far - 8.8 x 10^60 instants, to put it crudely and with apologies to Einstein.

Now let's count things. Defining only fundamental particles as things, in the standard model some have thrown a dart and come up 10^100 particles, one googol. That'll work for now. The vast number of permutations with this set of individual things is 10^100!. A scary big number, but still finite. Of course, if you count photons as things, photons vastly outnumber quarks and leptons by a factor of at least a billion. Fine; let's make it 10^209!. Then the number of instants in which things can have happened (8.8 x 10^60) multiplied by the possible combination of things in each instant (10^209!) is the number of things that can have happened so far in the universe. Let's call this huge but still finite term Ω.

You may argue with the figures I've used or even the rather ham-handed back-of-the-envelope calculation here, but my point is that the number of things that can have happened so far is finite, and so is the number of things that can ever happen, whether you expect a Big Rip or a proton decay at some point 10^10^70 years from now. In fact the real number of things that can have happened up until this point must be much smaller than what I've proposed; every arrangement of those 10^209 elements is constrained by the previous arrangement as a result of things like the speed of light and the conservation of energy.

So now we have Ω - so what? What's interesting is that there must also be a number Ω + 1; a number which exceeds possible events x things to describe - a number that cannot refer to anything real. Yes, Ω will get larger as time goes on, but it will still be finite, and arithmetic will always allow Ω + 1. That's nothing new; examples abound of theoretical computations that could not be completed before the expected decay of protons, even with the resources of the entire universe's fundamental particles marshalled for the task. Many of them involve board games.

So this means that mathematics - even arithmetic - is richer than it needs to be to describe our impoverished universe, and that there exist numbers which are simultaneously logically valid but which can in principle never have meaning in physical reality. My intuition is that this has less to say about reality than it does about mathematics, which is a particularly effective form of language we are just in the early process of developing to understand the world.

Words to Live By

"...there is good and bad speculation, and this is not an unparalleled activity in science...Those scientists who have no taste for this sort of speculative enterprise will just have to stay in the trenches and do without it, while the rest of us risk embarrassing mistakes and have a lot of fun." - Dan Dennett